Small cap decouplings

Abstract

We develop a toolbox for proving decouplings into boxes with diameter smaller than the canonical scale. As an application of this new technique, we solve three problems for which earlier methods have failed. We start by verifying the small cap decoupling for the parabola. Then we find sharp estimates for exponential sums with small frequency separation on the moment curve in $\mathbb {R}^3$. This part of the work relies on recent improved Kakeya-type estimates for planar tubes, as well as on new multilinear incidence bounds for plates and planks. We also combine our method with the recent advance on the reverse square function estimate, in order to prove small cap decoupling into square-like caps for the two dimensional cone. The Appendix by Roger Heath-Brown contains an application of the new exponential sum estimates for the moment curve, to the Riemann zeta-function.

Appendix: An Improved Fourth Derivative Estimate for Exponential Sums (D.R. Heath-Brown, Mathematical Institute, Oxford)

In this appendix we will show how Theorem 3.3 may be applied to establish an improved version of the “fourth derivative estimate” for exponential sums. The classical van der Corput k-th derivative estimate (Titchmarsh [Tit86, Theorems 5.9, 5.11, & 5.13], for example), can be given as follows. Suppose that $k\ge 2$ is an integer, and let $f(x):[0,N]\rightarrow \mathbb {R}$ have a continuous k-th derivative on (0, N) with $0<\lambda _k\le f^{(k)}(x)\le A\lambda _k$. Then

$$\begin{aligned} \sum _{n\le N}e(f(n))\lesssim A^{2^{2-k}}N\lambda _k^{1/(2^k-2)} +N^{1-2^{2-k}}\lambda _k^{-1/(2^k-2)}, \end{aligned}$$

(67)

where the implied constant is independent of k.

By using the (essentially) optimal estimate for Vinogradov’s mean value, as proved by Bourgain, Demeter and Guth [BDG16], one can obtain an alternative bound

$$\begin{aligned} \sum _{n\le N}e(f(n)) \lesssim _{A,k,\varepsilon }N^{1+\varepsilon } (\lambda _k^{1/k(k-1)}+N^{-1/k(k-1)}+N^{-2/k(k-1)}\lambda _k^{-2/k^2(k-1)}), \end{aligned}$$

(68)

for any fixed $\varepsilon >0$ (see Heath-Brown [Hea17, Theorem 1]). In most situations this is superior to the classical estimate as soon as $k\ge 4$, and the object of this appendix is to show how Theorem 3.3 of the present paper allows one to produce a further improvement in the case $k=4$. The result we obtain is the following.

Theorem 12.1

Let $f(x):[0,N]\rightarrow \mathbb {R}$ have a continuous 4th derivative on (0, N) with $0<\lambda _4\le f^{(4)}(x)\le A\lambda _4$ for some constant $A\ge 1$. Write $\lambda _4=N^{-\varpi }$, and suppose that $N^{-2}\lesssim \lambda _4\lesssim N^{-1}$. Then

$$\begin{aligned} \sum _{n\le N}e(f(n)) \lesssim _{A,\varepsilon } N^{1-\varpi /(4\varpi +8)+\varepsilon }+N^{8/9+\varepsilon }, \end{aligned}$$

(69)

for any fixed $\varepsilon >0$.

In practice one would usually apply the third derivative bound when $\lambda _4\lesssim N^{-2}$, giving a stronger result than can be obtained from the fourth derivative estimates. When $k=4$ and $\lambda _4=N^{-\varpi }$ the bound (68) yields

$$\begin{aligned} \sum _{n\le N}e(f(n)) \lesssim _{A,\varepsilon } \left\{ \begin{array}{cc} N^{1-\varpi /12+\varepsilon }, &{} \lambda _4\gg N^{-1},\\ N^{11/12+\varepsilon }, &{} N^{-2}\lesssim \lambda _4\lesssim N^{-1},\end{array}\right. \end{aligned}$$

(70)

while (69) produces

$$\begin{aligned} \sum _{n\le N}e(f(n)) \lesssim _{A,\varepsilon } \left\{ \begin{array}{cc} N^{1-\varpi /(4\varpi +8)+\varepsilon }, &{} N^{-8/5}\lesssim \lambda _4\lesssim N^{-1},\\ N^{8/9+\varepsilon }, &{} N^{-2}\lesssim \lambda _4\lesssim N^{-8/5}. \end{array}\right. \end{aligned}$$

(71)

Thus we get a significant saving when $N^{-2}\lesssim \lambda _4\lesssim N^{-1}$.

As an application of Theorem 12.1 we will prove a bound for the Lindelöf $\mu (\sigma )$ function associated to the Riemann Zeta-function.

Theorem 12.2

We have

$$\begin{aligned} \zeta (\tfrac{11}{15}+it)\lesssim _{\varepsilon }(|t|+1)^{1/15+\varepsilon } \end{aligned}$$

for any fixed $\varepsilon >0$, so that $\mu (\tfrac{11}{15})\le \tfrac{1}{15}$.

Strictly speaking, we do not claim that this bound is new. Indeed given the plethora of published bounds and the convexity of $\mu (\sigma )$ it is not easy to say with confidence that a given result is new. Moreover one can make further small improvements on Theorem 12.2 by using exponent pairs to sharpen the application of the third derivative bound in the argument below, and by replacing the trivial bound (for small N) by the case $k=5$ of (68). However our main purpose with Theorem 12.2 is to demonstrate a neat bound coming directly from the new fourth derivative estimate.

The proof of Theorem 12.1 begins by following the argument from [Hea17, Section 2]. We assume that $k=4$, although the initial stages of the method work for arbitrary $k\ge 3$. We write $H=[(A\lambda _k)^{-1/k}]$ and for $\varvec{\alpha }\in [0,1]^{k-1}$ we define

$$\begin{aligned} \nu (\varvec{\alpha })=\#\{n\le N-H: ||f^{(j)}(n)/j!-\alpha _j||\le H^{-j} \text{ for } 1\le j\le k-1\}. \end{aligned}$$

If we set $\alpha ^*=f^{(k-1)}(0)/(k-1)!$ then whenever $\nu (\varvec{\alpha })\not =0$ we must have

$$\begin{aligned} |\alpha _{k-1}-\alpha ^*|\le \frac{\left| f^{(k-1)}(n)-f^{(k-1)}(0)\right| }{(k-1)!}+H^{1-k} \lesssim _{A,k} N\lambda _k+\lambda _k^{(k-1)/k} \end{aligned}$$

for some $n\le N-H$. If we write this as $|\alpha _{k-1}-\alpha ^*|\le \xi$, say, then in our situation we have $\xi \lesssim _A N\lambda _4$, since $\lambda _4\gg N^{-2}$. We may now replace Lemma 1 of [Hea17] by the estimate

$$\begin{aligned} \sum _{n\le N}e(f(n)) \lesssim _{A,\varepsilon } H+N^{1-1/s}\mathcal {N}^{1/2s} \left\{ H^{-2s+k(k-1)/2}J\right\} ^{1/2s}, \end{aligned}$$

with

$$\begin{aligned} \mathcal {N}=\#\left\{ m,n\le N: \left| \left| \frac{f^{(j)}(m)}{j!}-\frac{f^{(j)}(n)}{j!}\right| \right| \le 2H^{-j} \text{ for } 1\le j\le k-1\right\} \end{aligned}$$

and

$$\begin{aligned} J=\int _0^1\ldots \int _0^1\int _{\alpha ^*-\xi }^{\alpha ^*+\xi } \left| \sum _{n\le x}e(\alpha _1 n+\cdots +\alpha _{k-1} n^{k-1})\right| ^{2s}d\alpha _{k-1} d\alpha _{k-2}\ldots d\alpha _1 \end{aligned}$$

for some $x\le H$. Note that the argument of [Hea17, Section 2] works for any real $s\ge 1$.

We now specialize to $k=4$, and plan to apply Theorem 3.3. As remarked in connection with Theorem 3.3 the proof allows us to replace the range $[0,N^{-\beta }]$ by any interval of length $N^{-\beta }$. What is less clear is whether the estimate of the theorem holds uniformly with respect to $\beta$. To clarify this point we suppose that the theorem yields the bounds

$$\begin{aligned}&\int _{[0,1]^2\times [\tau ,\tau +N^{-\beta }]}|\sum _{n\le N} e(\alpha _1 n+\alpha _2n^2 +\alpha _3n^3)|^{12-2\beta }d\alpha _1 d\alpha _2 d\alpha _3\\&\quad \le \, C(\varepsilon ,\beta )N^{6-2\beta +\varepsilon }&, \end{aligned}$$

for $0\le \beta \le \tfrac{3}{2}$, uniformly in $\tau$. Set $R=\lceil \tfrac{3}{2}\varepsilon \rceil$ and $r=\lceil \tfrac{2}{3} R\beta \rceil$. If we then write $\beta _r=3r/2R$ it follows that $0\le \beta _r\le 3/2$ and $\beta _r-\varepsilon <\beta \le \beta _r$. We now observe firstly that

$$\begin{aligned} |\sum _{n\le N}e(\alpha _1 n+\alpha _2n^2 +\alpha _3n^3)|^{12-2\beta } \le N^{2(\beta _r-\beta )} |\sum _{n\le N}e(\alpha _1 n+\alpha _2n^2 +\alpha _3n^3)|^{12-2\beta _r}, \end{aligned}$$

and secondly that the interval $[\tau ,\tau +N^{-\beta }]$ can be covered by at most $N^{\varepsilon }$ intervals of length $N^{-\beta _r}$. Thus

$$\begin{aligned}&\int _{[0,1]^2\times [\tau ,\tau +N^{-\beta }]}|\sum _{n\le N} e(\alpha _1 n+\alpha _2n^2 +\alpha _3n^3)|^{12-2\beta }d\alpha _1 d\alpha _2 d\alpha _3\\&\quad \le N^{3\varepsilon }\sup _\sigma \int _{[0,1]^2\times [\sigma ,\sigma +N^{-\beta _r}]}|\sum _{n\le N} e(\alpha _1 n+\alpha _2n^2 +\alpha _3n^3)|^{12-2\beta _r}d\alpha _1 d\alpha _2 d\alpha _3\\&\quad \le C(\varepsilon ,\beta _r)N^{6-2\beta _r+4\varepsilon }\\&\quad \le C(\varepsilon ,\beta _r)N^{6-2\beta +4\varepsilon }. \end{aligned}$$

We therefore see that Theorem 3.3 holds (with $\varepsilon$ replaced by $4\varepsilon$) with implied constant $C(\varepsilon )=\max _{r\le R}C(\varepsilon ,\beta _r)$ depending only on $\varepsilon$.

We proceed to apply this uniform version of Theorem 3.3. We have assumed that $\Lambda _4\le cN^{-1}$ for some constant c. With this in mind we define $\beta$ by the relation

$$\begin{aligned} H^{\beta }=\min \left\{ c(N\lambda _4)^{-1}\,,\,H^{3/2}\right\} . \end{aligned}$$

We then have $0\le \beta \le \tfrac{3}{2}$ as required. Moreover,

$$\begin{aligned} H=[(A\lambda _4)^{-1/4}]=N^{\varpi /4+O(1/\log N)} \end{aligned}$$

and $1\lesssim \varpi \lesssim 1$, whence $N=H^{4/\varpi +O(1/\log N)}$ and

$$\begin{aligned} c(N\lambda _4)^{-1}=H^{4(\varpi -1)/\varpi +O(1/\log N)}. \end{aligned}$$

We therefore see that

$$\begin{aligned} \beta =\min \left\{ \frac{4(\varpi -1)}{\varpi }\,,\,\frac{3}{2}\right\} +O\left( \frac{1}{\log N}\right) . \end{aligned}$$

Since $\xi \lesssim _A N\lambda _4\lesssim H^{-\beta }$ we can cover the range $[\alpha ^*-\xi ,\alpha ^*+\xi ]$ with $O_A(1)$ intervals of length $H^{-\beta }$. It then follows on taking $s=6-\beta$ that

$$\begin{aligned} J\lesssim _{\varepsilon } H^{2s-6+\varepsilon }. \end{aligned}$$

Moreover [Hea17, Lemma 3] yields $\mathcal {N}\lesssim _{A,\varepsilon }N^{1+\varepsilon }$ when $N^{-2}\lesssim \lambda _4\lesssim N^{-1}$. We therefore conclude that

$$\begin{aligned} \sum _{n\le N}e(f(n)) \lesssim _{A,\varepsilon } \lambda _4^{-1/4}+N^{1-1/2s+\varepsilon }. \end{aligned}$$

However

$$\begin{aligned} \frac{1}{2s}=\frac{1}{12-2\beta }= \min \left\{ \frac{\varpi }{4\varpi +8}\,,\,\frac{1}{9}\right\} +O((\log N)^{-1}), \end{aligned}$$

whence

$$\begin{aligned} \sum _{n\le N}e(f(n)) \lesssim _{A,\varepsilon } N^{\varpi /4}+N^{1-\varpi /(4\varpi +8)+\varepsilon }+N^{8/9+\varepsilon }. \end{aligned}$$

Theorem 12.1 then follows.

To deduce Theorem 12.2 it suffices by the approximate functional equation (see Chapter 2 in [GK91] or [Tit86]) to show that

$$\begin{aligned} \sum _{N<n\le 2N}n^{it}\lesssim _{\varepsilon }N^{11/15}t^{1/15+\varepsilon } \end{aligned}$$

for $N\le t^{1/2}$ and any fixed $\varepsilon >0$. The bound is trivial for $N\le t^{1/4}$, and so we focus on the remaining range $t^{1/4}\le N\le t^{1/2}$. When $f(x)=t(\log x)/2\pi$ one may apply the third derivative estimate, taking $\lambda _3$ to have order $tN^{-3}$. The bound (67) then shows that

$$\begin{aligned} \sum _{N<n\le 2N}n^{it}\lesssim N^{1/2}t^{1/6}+Nt^{-1/6}. \end{aligned}$$

This gives a satisfactory bound $O(N^{11/15}t^{1/15})$ when $t^{3/7}\le N\le t^{1/2}$. For the remaining range $t^{1/4}\le N\le t^{3/7}$ we use our various fourth derivative estimates, with $\lambda _4$ of order $tN^{-4}$. When $t^{1/4}\le N\le t^{1/3}$ the bound (70) yields

$$\begin{aligned} \sum _{N<n\le 2N}n^{it}\lesssim _{\varepsilon } N^{1+\varepsilon }\lambda _4^{1/12}\lesssim N^{2/3+\varepsilon }t^{1/12} \lesssim N^{11/15+\varepsilon }t^{1/15}. \end{aligned}$$

For $t^{5/12}\le N\le t^{3/7}$ we have $N^{-5/3}\lesssim \lambda _4\lesssim N^{-8/5}$ so that (71) produces an estimate

$$\begin{aligned} \sum _{N<n\le 2N}n^{it}\lesssim _{\varepsilon }N^{8/9+\varepsilon }\le N^{11/15+\varepsilon }t^{1/15}. \end{aligned}$$

Finally, when $t^{1/3}\le N\le t^{5/12}$ we find that $N^{-8/5}\lesssim \lambda _4\lesssim N^{-1}$. In this case (71) shows that

$$\begin{aligned} \sum _{N<n\le 2N}n^{it}\lesssim _{\varepsilon } N^{1-\varpi /(4\varpi +8)+\varepsilon }. \end{aligned}$$

If we write $t=N^{\tau }$ we will have $\tfrac{12}{5}\le \tau \le 3$, and $\varpi =4-\tau +O(1/\log N)$. It therefore suffices to show that

$$\begin{aligned} 1-\frac{4-\tau }{24-4\tau }\le \frac{11}{15}+\frac{\tau }{15} \end{aligned}$$

for $\tfrac{12}{5}\le \tau \le 3$, and this is readily verified, completing the proof of Theorem 12.2. The reader will note that the critical case is that in which $\lambda _4$ is of order $N^{-5/3}$.

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